Dry Powder Sometimes Explodes: Contracting on Fund Size and Investment Duration in Private Equity

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1 Dry Powder Sometimes Explodes: Contracting on Fund Size and Investment Duration in Private Equity Dawei Fang Department of Economics, University of Gothenburg 15 th February, 2015 Abstract: Private equity (PE) funds typically invest in a portfolio of projects and PE managers have considerable discretion over investment timing. Empirical studies find evidence of managerial opportunism and question whether the existing PE fund structure is optimal. We tackle this question by examining the interaction between investment pooling and investment timing. Pooling helps design more efficient contracts but gives fund managers incentives to make investments sequentially and gamble for resurrection when early investments fail. Restricting investment timing by shortening investment duration contracts off gambling for resurrection, but it improves efficiency only if managerial compensation sufficiently caps managers upside sharing, which may force managers to concede rents to competitive investors. When projects have high profitability potential, the equilibrium fund structure coincides with that in practice and results in managerial opportunism found in the empirical literature. Our results imply that the existing PE fund structure is optimal for fund managers but not society. Contact dawei.fang@cff.gu.se. I thank Alexander Guembel, Ulrich Hege, Tim Jenkinson, Alan Morrison, Thomas Noe, Ludovic Phalippou, John Quah, David Robinson, Linus Siming, and Anjan Thakor for helpful comments, and seminar participants at EFMA 2014, University of Gothenburg, University of Oxford, and Queen Mary University of London. Any remaining errors are my responsibility.

2 1 Introduction Private equity (PE) funds have a limited partnership structure with a finite lifespan, usually years (Robinson and Sensoy (2013)). The life of a PE fund is contractually partitioned into an investment period (usually the first 5 years of the fund) and a harvesting period. New investments are limited to the investment period (Metrick and Yasuda (2010)), during which the management team of the PE fund, who serves as the general partner (GP), seeks new investment opportunities and invests the capital committed by the limited partners (LPs) in a portfolio of projects. During the harvesting period, the GP manages and eventually exits from those investments to realize capital gains. GP compensation generally includes an option-like performance component, known as carried interest (or carry), which usually amounts to 20% of the overall performance of the fund above a pre-specified hurdle rate. 1 This incentive scheme aims to resolve the potential agency problems. However, as pointed out by Arcot, Fluck, Gaspar, and Hege (2014), for funds late in the investment period with substantial dry powder (unspent capital), the same (incentive) contract creates adverse incentives to window dress. Recent empirical studies find evidence of GP opportunism in the late phase of the investment period. For example, Dass, Hsu, Nanda, and Wang (2012), Arcot, Fluck, Gaspar, and Hege (2014), and Braun and Schmidt (2014) all find that a PE fund s late investments significantly underperform its early investments. Moreover, Degeorge, Martin, and Phalippou (2013) find that the secondary buyouts (SBOs) made in the late phase of the investment period of a PE fund not only underperform the SBOs made in the early phase but have negative NPVs on average. These findings suggest that the GP with substantial dry powder close to the end of the investment period has an incentive to do deals that are not in the interest of LPs. This naturally raises three questions. First, what drives such a phenomenon? Second, is there an alternative PE contract that can possibly alleviate GP opportunism? For example, Degeorge, Martin, and Phalippou (2013) suggest that LPs might benefit from contractually capping the percentage of a fund that the GP may invest in late SBOs. Third, if there exists a more efficient PE contract, how to explain the use of a suboptimal contract in practice? In this paper, we tackle these three questions based on a simple agency model, in which we endogenize GP compensation, fund size, and the length of the investment period of the partnership. In our model, to weed out the unskilled fly-by-night operators, the fund gives the GP a positive payoff only if the GP creates value for the fund. Given 1 See Sahlman (1990), Gompers and Lerner (1999), Phalippou (2007), Cumming and Johan (2009), and Metrick and Yasuda (2011) for a discussion of the PE partnership structure and GP compensation. 1

3 this option-like characteristic of GP compensation, the GP bears little of the downside risk and thus has an inclination toward risk shifting. We show that, if the partnership specifies a long investment period, the GP has an incentive to strategically time his investment decisions: he undertakes better quality projects in the early phase of the investment period and postpones his investment decisions regarding the worse quality projects until he accesses private information on the progress of his earlier investments. This strategic timing behavior enables the GP, in the middle of the investment period, to better predict the fund s final performance. Since GP compensation bases on the fund s final performance, the interim private information on fund performance is useful for the GP to game the compensation scheme in the late phase of the investment period. Particularly, if he anticipates the success of his earlier investments, he will pass up the worse quality projects, while if he foresees the failure of his earlier investments, he will gamble for resurrection by taking those bad deals. These results indicate that the calloption-like GP compensation and the long investment period without harsh restrictions on late investments are the key drivers for observed GP opportunism. To improve efficiency, the contract must restrict the GP s late investments by, for example, capping the amounts the GP can invest in the late phase, which is suggested by Degeorge, Martin, and Phalippou (2013), or shortening the investment period. However, such a restriction is only necessary but not sufficient to better discipline the GP. This is because, although such a restriction alleviates GP opportunism in the late phase of the investment period, it can exacerbate GP opportunism in the early phase. We show that, when project profitability potential is high and when the GP has a considerable share in the upside, restricting late investments induces the GP to gamble more aggressively for the upside in the early phase and causes more overinvestment in total. Thus, when project profitability potential is high, socially optimal contracting requires restricting both the GP s late investments and the GP s upside sharing, which may force the GP to concede rents to LPs. When project profitability potential is sufficiently high, upside concession results in large rent concession, which makes this socially optimal contract privately suboptimal for the GP to use. In this case, the GP s privately optimal contract imposes no late investment restriction and gives the GP a call-option-like security. Although this contract is not socially optimal, it is rent-saving and is more efficient than many other alternative financing arrangements. Since, in practice, PE funds typically invest in projects with high profitability potential, our model implies that the socially optimal contract is costly for the GP to use because of rent concession. Thus, our model offers an explanation for both the existing PE fund structure and the existence of GP opportunism found in the empirical studies. We further show that, when the socially optimal contract is not used, restricting fund 2

4 size can serve as an alternative but less efficient way to alleviate GP opportunism. A small fund minimizes overinvestment but also creates an underinvestment problem: the GP has to pass up some good investment opportunities if he encounters a sufficiently large number of good projects. Thus, fund size is restricted only if underinvestment does not create much inefficiency, which is likely the case when economic conditions are bad so that the GP has low chance of facing many good investment opportunities. Apart from the implications on PE fund structures, our model also contains a number of predictions for GP behavior, PE fund size and performance, and LP earnings. We list some of these predictions as follows: i. The GP takes more (less) risks if the fund has a lower (higher) cumulative performance. This prediction is consistent with the evidence in Ljungqvist, Richardson, and Wolfenzon (2008) and Barrot (2014) that, following periods of good (bad) performance, GPs become more (less) conservative. ii. The GP with more dry powder in the late phase of the investment period is more likely to take bad deals. This prediction receives empirical support from Degeorge, Martin, and Phalippou (2013) and Arcot, Fluck, Gaspar, and Hege (2014). iii. Projects undertaken earlier by a PE fund are more profitable than those undertaken later by the same fund. This coincides with the empirical findings of Dass, Hsu, Nanda, and Wang (2012), Degeorge, Martin, and Phalippou (2013), Arcot, Fluck, Gaspar, and Hege (2014), and Braun and Schmidt (2014) that a PE fund s late investments significantly underperform its early investments. iv. The funds which invest quicker outperform the funds with lower investment speed. This prediction receives support from the evidence in Giot, Hege, and Schwienbacher (2014) that experienced funds invest quicker than novice funds and outperform novice funds at a highly significant margin. v. LPs do not earn risk-adjusted excess returns from investing in PE funds, which is consistent with the findings in Phalippou (2014) and Sorensen, Wang, and Yang (2014). vi. PE fund size is procyclical. This is a testable prediction. Existing literature finds evidence of procyclicality of PE fundraising on the industry level. For example, Kaplan and Strömberg (2009) find that PE fundraising booms and busts are strongly correlated with public equity booms and busts. However, to our knowledge, procyclicality of fund size has not been tested. 3

5 To our knowledge, this is the first paper that studies the investment timing issue under the principal-agent framework when investment pooling is used. The theoretical literature on investment pooling typically focuses on how pooling can help design incentive compatible contracts, through the channels of risk diversification (Diamond (1984)) or inside wealth creation (Laux (2001) and Tirole (2006)). Although our paper confirms the benefits of investment pooling, it focuses on the interaction of investment pooling and investment timing. Our paper shows that, given investment pooling, the agent s investment decisions on part of the projects can be affected by the progress of the other part and the agent has an incentive to game the timing of investment decisions to exploit the contractual provisions. Such an incentive must be taken into account when designing optimal contract. The literature on investment timing typically takes a real options approach that emphasizes the value of waiting to invest under uncertainty. This literature usually adopts a decision-theoretic framework by examining the optimal investment/divestment timing rules with no agency problems involved (see Pindyck (1991) for an excellent survey). The effects of moral hazard on investment timing decisions is studied in Grenadier and Wang (2005), with a real options model in which the agent manages a single project. The key difference between our model and the real options models is that the NPV of a project is fixed in our model whereas in real options models, it generally follows a stochastic process. Thus, in contrast to real options models, in our model, the manager s incentive to wait is not driven by project quality fluctuation but by agency conflicts and investment pooling. This paper also relates to the theoretical literature analyzing PE fund structures. Kandel, Leshchinskii, and Yuklea (2011) argue that the finite lifespan of the partnership can induce the GP to make inefficient investments in risky projects. Their analysis is based on the exogenously specified GP compensation structure and a predetermined investment period of the fund. In contrast, this paper endogenizes both GP compensation and the length of the investment period and shows that inefficient fund structures can arise in equilibrium. Inderst, Mueller, and Münnich (2007) argue that investment pooling, combined with capital rationing, helps the GP commit to efficient liquidation decisions. However, in this paper, capital rationing never minimizes agency conflicts. Axelson, Strömberg, and Weisbach (2009) study the GP s financing of a portfolio of projects. They show that, in the pure ex ante financing setting, it is never optimal to make the GP capital-constrained, and they find that the optimal financing structure is a combination of ex ante pooled financing and ex post deal-by-deal financing. In contrast, this paper shows that, in the pure ex ante financing setting, although pooled financing can increase efficiency, the GP may prefer having the fund capital-constrained in order to save rents. Fulghieri and Sevilir (2009) study the optimal size and scope of the 4

6 fund s portfolio by focusing on the double-sided moral hazard problem between the GP and the entrepreneurs of the projects. They show that the GP finds it optimal to limit portfolio size when projects have high profitability potential and the GP expands the portfolio size only when project fundamentals are more moderate. In contrast, this paper abstracts from the effort-incentive problem and focuses on the GP-LP relationship rather than the GP-entrepreneur relationship. This paper finds that, even with exogenously fixed project quality, heterogeneous fund sizes can arise in equilibrium and the GP deliberately limits fund size only when project profitability potential is not too high. The remainder of the paper is structured as follows. Section 2 describes the model. Section 3 analyzes the socially optimal financing arrangement and shows that this arrangement may force the GP to concede rents to LPs. Section 4 analyzes alternative financing arrangements, which are less efficient but rent-saving. Section 5 determines the equilibrium financing arrangement. Section 6 provides the implications of the model. Section 7 concludes. All the proofs are relegated to the Appendix. 2 The model 2.1 The players There are three types of players in the model: a GP, investors who are perfectly competitive, and fly-by-night operators. 2 All players are risk-neutral, with no discounting of the future, and have access to a storage technology yielding the risk-free interest rate, which is normalized to zero. 2.2 The projects The model has four dates (time 0, 1, 2, and 3). At time 0, all the players anticipate that the GP will be endowed with two projects at time 1 and they have the common prior that the quality of each project will be independently drawn from the same distribution, with probability λ (0,1) of being good (G) and probability 1 λ of being bad (B). 3 At time 1, two projects arrive. The quality of each project is privately observable to the GP, who possesses certain expertise that normal investors lack. Each project requires an investment of I to be undertaken. To undertake a project, the GP must invest I in the project at either time 1 or time 2; otherwise, the investment opportunity disappears. To 2 We use investors and LPs interchangeably throughout the paper. 3 There is no change in the qualitative nature of the result if we assume positive project quality correlation. This extension is available from the author upon request. 5

7 focus on the investment timing effect on investment efficiency through the GP s opportunistic behavior but not through the change in project quality, we assume no change in a project s quality no matter whether it is undertaken at time 1 or 2. 4 Particularly, we assume that a good project, if undertaken at either time 1 or 2, will succeed, producing cash flow R > I, with certainty at time 3. A bad project, if undertaken at either time 1 or 2, will succeed with some probability, i.e., it will produce, at time 3, R with probability p (0,1) and 0 with probability 1 p. We assume that bad projects are negative NPV investments, i.e., pr < I. To simply the analysis, we further assume that R > 2I. (1) Thus, the costs of investing in both projects can be covered when at least one investment succeeds. 5 Given that pr < I, condition (1) further implies that p < 1 2. To give the GP incentives to postpone part of his investment decisions, we assume that the progress of investments made at time 1 is privately observable to the GP at time 2. To deliver the key insight without getting into unnecessary complications, we assume that, at time 2, the GP knows whether the investments made at time 1 will eventually succeed or fail. Note that, if the GP aims to maximize efficiency, then such interim information is useless, because a project s cash flow distribution is independent of the outcome of the other project. However, if there is misalignment between the GP s interests and social interests, this interim information can be useful for the GP s investment decision at time 2, as we will see later on. 2.3 The GP s problem The GP has no money of his own and has to raise money from investors. 6 We focus on ex post information asymmetry regarding project quality by assuming that the GP has to raise funds by establishing a partnership with investors at time 0 (when the GP has no superior information on project quality). This assumption can be justified on the 4 Introducing project quality deterioration caused by investment delay will not change the qualitative nature of our result so long as the reduction in the probability of success for a bad project caused by delay is sufficiently small. 5 There is no change in the qualitative nature of the result if we extend the analysis to I < R 2I. This extension is available from the author upon request. 6 Although PE funds may have capital contribution provisions that require the GP to contribute 1% of the committed capital, the GP is frequently permitted to borrow money from the fund to meet his capital contribution requirements (see Harris (2010)). 6

8 grounds that PE fundraising is time consuming. The partnership lasts for three periods and is dissolved at time 3 when all cash flows are realized. At time 0, to raise funds, the GP offers investors a contract that specifies fund size, the length of the investment period, and GP compensation. Fund size is determined by investors committed capital, denoted by K. Because there are at most two good projects and because, in our model, having a financial slack does not help contracting, without loss of generality, we restrict the choice of K to K {I,2I}. When K = I (K = 2I), the fund size is small (big) and the GP can undertake at most one project (two projects). We denote the length of the investment period by l {1,2}, where l = 1 (l = 2) refers to a short (long) investment period, including time 1 only (both time 1 and 2). Note that new investments are limited to the investment period, so the GP is allowed to make new investments at time 2 only when the investment period is long, i.e., when l = 2. Let s(x) denote the GP security, which maps the final cash flow x to the GP s payoff and satisfies the following limited liability and monotonicity conditions: Limited Liability (LL): 0 s(x) x for all x 0; Monotonicity (M): s(x) and x s(x) are nondecreasing in x. The monotonicity condition is quite common in the financial literature on security design, e.g., Harris and Raviv (1989) and Nachman and Noe (1994). This condition contracts off the GP s moral hazard of increasing his own payoff by secretly borrowing money from a third party or by burning money. Given s, the investor security is automatically determined by x s(x). To introduce agency conflicts in a simple way (otherwise the two parties interests can be perfectly aligned by using an equity contract), we follow Axelson, Strömberg, and Weisbach (2009) by introducing an ex ante adverse selection problem, where flyby-night operators play a role. We assume that there is an infinite supply of fly-by-night operators who cannot bring success to any project but can store money at the riskfree interest rate. At time 0, without contracts as a signalling device, investors cannot distinguish capable GPs from unskilled fly-by-night operators. The ex ante contracting must resolve this adverse selection problem, because if fly-by-night operators have an incentive to mimic capable GPs by offering the same contract, given the infinite supply of fly-by-night operators, the probability that investors meet a fly-by-night operator would be one, in which case investors must lose money. Because fly-by-night operators can store money at the risk-free interest rate, to prevent them from mimicking, the contract has to ensure that they earn nothing by storing money. Thus, the GP security must satisfy the following condition: No-Profit-No-GP-Earning (NP): for raised capital K, s(x) = 0 whenever x K. Condition (NP) makes the GP bear little of the downside risk and thus induces the 7

9 GP s risk-shifting tendency. This condition is also consistent with GP compensation in practice: GPs earn no performance-based compensation before LPs are paid in full (see Sahlman (1990)). The GP s problem is to maximize his expected payoff by offering investors, at time 0, a contract (K, l, s) that specifies fund size, investment duration, and GP compensation. Investors accept the contract and provide the GP with K if they anticipate that they can at least break even. Figure 1 illustrates the timeline of the model. Time 0 Time 1 Time 2 Time 3 GP establishes partnership Partnership contracting: fund size investment period GP security Projects arrive GP observes quality GP makes investment decisions GP observes the progress of time 1 s investments If GP has dry powder and investment period is long, GP decides whether to invest dry powder in more projects All cash flows are realized Partnership is dissolved Figure 1: The timeline of the model We make one final assumption that specifies the GP s preference when two or more actions give him the same payoff. We assume that the GP has a lexicographic ordering of preferences over actions that give him the same payoff: (a) he prefers those that enable him to possibly have a successful project at time 3 to those that give him zero probability of having a successful project and (b) among the actions that give him positive probability of having a successful project, he prefers the one that produces higher investment surplus. This assumption is not crucial for the main results. It is mainly used to simplify the expression for the mathematical results in the borderline cases. Under this assumption, if the GP faces two bad projects and if we suppose that the GP chooses between undertaking 0, 1, and 2 of them with all the three choices delivering zero payoff to him, he will choose to undertake 1 of them. 3 The socially optimal financing arrangement 3.1 Conditions Since all the ex post efficiency loss will be borne by the GP ex ante, it is natural to start our analysis by investigating the optimal contract that minimizes inefficiency. At time 1, there are three states of the world, distinguished by the combination of two projects quality: (1) state GG, in which both projects are good, (2) state GB, in which one 8

10 project is good while the other is bad, 7 and (3) state BB, in which both projects are bad. Because of condition (NP), the first-best world, in which the GP invests in all good but no bad projects in all the three states, is unattainable: in state BB, the GP always has an incentive to invest in at least one bad project, since otherwise he earns nothing. Thus, it is impossible to contract off the efficiency loss from undertaking one bad project in state BB. It is thus natural to examine the second best, i.e., the financing arrangement that induces the GP to i. invest in two good projects in state GG, ii. invest in only the good project in state GB, iii. and invest in only one bad project in state BB. The investment surplus from the above prescribed investment behavior equals π s = λ 2 (2R 2I) + 2λ(1 λ)(r I) + (1 λ) 2 (pr I) (2) = ( 2λ + (1 λ) 2 p ) R (1 + λ 2 )I. In expression (2), 2R 2I is the surplus from investing in two good projects in state GG, which occurs with probability λ 2 ; R I is the surplus from investing in only the good project in state GB, which occurs with probability 2λ(1 λ); pr I is the surplus from investing in only one bad project in state BB, which occurs with probability (1 λ) 2. Financing is feasible only if the surplus expressed in (2) is nonnegative. This condition is equivalent to R/I 1 + λ 2 2λ + (1 λ) 2 p = r 0(p,λ). (3) We think of R/I as representing a project s profitability potential. Thus, financing is feasible only if the project s profitability potential is greater than the threshold, r 0, defined in (3). In what follows, we assume that condition (3) holds. We call the financing arrangements that induce the GP to follow the above prescribed investment behavior socially optimal arrangements. Note that, socially optimal arrangements must satisfy the following three conditions: Unconstrained Capital: sufficient funds raised for investing in two projects; Cross-Pledging: putting the two projects together under one roof; Short Investment Period: new investments are limited to time 1. The first condition unconstrained capital is necessary to contract off underinvestment; it requires that the GP raises 2I at time 0. The second condition cross-pledging is necessary to induce the GP to abandon a bad project, since if the GP instead estab- 7 Given that the two projects are identical, it is irrelevant which of the two projects is good. We therefore treat these two separate but symmetric cases effectively as a single case. 9

11 lishes two funds, each used to finance one project, given that each fund s contract has to satisfy condition (NP), the GP will never abandon any bad project. Given unconstrained capital and the use of cross-pledging, the third condition short investment period is necessary to minimize overinvestment. To see this, note that, if the contract specifies a long investment period, in state BB, the GP has an incentive to invest in one bad project at time 1 and, if he foresees the failure of this investment at time 2, he will gamble for resurrection by undertaking the other bad project at time 2. 8 The above analysis implies that a socially optimal arrangement must specify large fund size and a short investment period, i.e., K = 2I and l = 1. Conditioned on K = 2I and l = 1, to minimize agency conflicts, the GP security, s, must satisfy the following incentive-compatibility conditions: 9 s(r + I) ps(2r) + (1 p)s(r), (IC GB ) ps(r + I) p 2 s(2r) + 2p(1 p)s(r). (IC BB ) The left-hand side of (IC GB ) is the GP s expected payoff in state GB if he only undertakes the good project. The right-hand side of (IC GB ) is the GP s expected payoff in state GB if he undertakes both projects. When (IC GB ) holds, the GP only undertakes the good project in state GB. The left-hand side of (IC BB ) is the GP s expected payoff in state BB if he only undertakes one bad project. The right-hand side of (IC BB ) is the GP s expected payoff in state BB if he undertakes both projects. When (IC BB ) holds, conditioned on l = 1, the GP only undertakes one bad project in state BB. It is noticeable that (IC BB ) implies (IC GB ). Thus, a contract is socially optimal if and only if it is in the form of (K = 2I,l = 1,s) with s satisfying (IC BB ) and conditions (LL), (M), and (NP) specified in Section 2.3. Since our aim is to derive the equilibrium financing arrangement, that is, given a perfectly competitive capital market, the contract that maximizes the GP s payoff, it is natural for us to examine, among all the socially optimal contracts, the contract that maximizes the GP s payoff. Denote this contract by C s and denote by s s the associated GP security. The above analysis implies that C s = (K = 2I,l = 1,s s). In the next part of this section, we solve for s s. 8 Admittedly, our specification of the GP s preferences over actions that deliver him the same payoff plays a role here. If we instead assume that, among the actions that give the GP the same payoff, the GP always prefers the one that maximizes investment efficiency, then gambling for resurrection can be contracted off by having s(r) = 0. However, even in this case, imposing a short investment period will help minimize overinvestment. This is because, with a short investment period, we no longer require s(r) = 0 to design the incentive-compatible contract and, hence, the security design problem is relaxed. 9 Since we have assumed that good projects will succeed for sure, the incentive-compatibility conditions that induce the GP to undertake good projects are implied by the monotonicity condition (M). 10

12 3.2 Security design With cross-pledging, the final cash flow x can potentially take on one of the six different values, i.e., x {0,I,2I,R,I + R,2R}. The GP security, s, must specify the GP s payoff for each of these six cash flow realizations, among which, by condition (NP), s(0) = s(i) = s(2i) = 0. Thus, s s must solve the following program: maxe[s(x)] = λ 2 s(2r) + [ 2λ(1 λ) + (1 λ) 2 p ] s(r + I), s (P) subject to (LL), (M), (NP), (IC BB ), and the following investors break-even condition E[s(x)] π s, (BE) where πs is given by equation (2). There are two possible payoffs to the GP in the maximand. The first payoff, s(2r), occurs only in state GG. The second payoff, s(r + I), occurs both in state BB when the bad project undertaken succeeds and in state GB. 10 For investors to break even, the GP s payoff must not exceed the investment surplus, given by (2). To solve the problem (P), we first solve the following relaxed problem: maxe[s(x)] = λ 2 s(2r) + [ 2λ(1 λ) + (1 λ) 2 p ] s(r + I), s (P ) subject to (LL), (M), (NP), and (IC BB ). Denote the solution to the above relaxed problem by s o. The only difference between the relaxed problem and the primal problem is the exclusion/inclusion of condition (BE). (BE) is investors break-even condition which imposes the investment surplus as an upper bound on the objective function. Thus, if the maximal value of the relaxed problem falls below the investment surplus, the solution to the primal problem equals the solution to the relaxed problem, while if otherwise, the solution to the primal problem equals a scale transformation of the solution to the relaxed problem. This result is formally presented in the following lemma. Lemma 1. Suppose s s and s o represent the solutions to the primal problem (P) and to the relaxed problem (P ), respectively. Then [ s s(x) = s o π ] s (x)min 1, E[s o (x)] x 0, (4) 10 If in state BB, the undertaken bad project fails, the final cash flow is I. By condition (NP), s(i) equals zero and is hence missing in the maximand. 11

13 where E[s o (x)] represents the maximal value of the relaxed problem (P ) and π s is the investment surplus given by (2). Our first proposition presents the expressions for s o and E[s o (x)]. Proposition 1. The solution to the relaxed problem (P), s o, and the associated maximal value E[s o (x)] are as follows: s o (x) = 0 for all x 2I, and i. if R/I 3 2p, then s o (R) = R 2I, s o (R + I) = R I, and s o (2R) = 2R 2I, in which case, ii. if 3 2p < R/I 3, then a. if λ λ, where E[s o (x)] = ( 2λ + (1 λ) 2 p ) (R I); (5) λ = p 2 2p + p p 2 1 p 2, (6) then s o (R) = R 2I, s o (R+I) = R I, and s o (2R) = R I +(3I R)(1 p)/p, in which case E[s o (x)] = (2λ + (1 λ) 2 p λ 2 ) R p (2λ + 2λ 2 + (1 λ) 2 p 3λ 2 p ) I, b. while if λ > λ, then s o (R) = (I pr)/(1 p), s o (R+I) = (I pr)/(1 p)+i, and s o (2R) = (I pr)/(1 p) + R, in which case, ( E[s o (x)] = 1 + λ 2 + (1 λ) 2 p 1 ) ( R+ (1 λ)(3λ 1) + (1 λ) 2 p + 1 ) I; 1 p 1 p (8) iii. if R/I > 3, then a. if λ λ, then s o (R) = I and s o (R + I) = s o (2R) = 2I, in which case E[s o (x)] = ( 4λ 2λ 2 + 2(1 λ) 2 p ) I, (9) (7) b. while if λ > λ, the expressions for s o and E[s o (x)] are the same as those presented in (iib). Proposition 1 has implications on socially optimal security design. When projects profitability potential, represented by R/I, is low (below the threshold 3 2p), part (i) in Proposition 1 implies that s o has all the marginal cash flows in the region [2I,2R] pledged to the GP. Thus, s o is a call option with strike price equal to LPs committed capital. Given that, by Lemma 1, s s is a scale transformation of s o, s s is a call-optionlike security with strike price equal to LPs committed capital. This case is illustrated in Panel A of Figure 2. 12

14 When projects profitability potential is not too low (above the threshold 3 2p), an incentive-compatible security must not have all the marginal cash flows in the region [2I,2R] pledged to the GP. In this case, if the projects have high probability of being good (λ > λ), parts (iib) and (iiib) in Proposition 1 imply that s o is a call option with strike price between 2I and R. Thus, given that s s is a scale transformation of s o, s s is a call-option-like security with strike price between 2I and R. This case is illustrated in Panel B of Figure 2. If projects have low probability of being good (λ λ), part (iia) in Proposition 1 implies that s o must have some marginal cash flows in the region [R + I, 2R] pledged to the LPs when projects have medium profitability potential (between 3 2p and 3), while part (iiia) implies that s o must have all the marginal cash flows in the region [R + I,2R] and some marginal cash flows in the region [2I,R] pledged to the LPs when projects have high profitability potential (above 3). This implies, given that s s is a scale transformation of s o, that in both cases, s s must exhibit concavity in the high cash flow region. These two cases are illustrated in Panels C and D in Figure 2. Proposition 1 also presents the expression for E[s o (x)], which is the GP s maximal payoff by using a socially optimal contract with LPs break-even condition omitted. Including LPs break-even condition just imposes the investment surplus, πs, as an upper bound on the GP s payoff. Thus, with LPs break-even condition included, the GP s payoff by using the socially optimal contract equals the minimum of E[s o (x)] and πs. This result is presented in the next proposition. Proposition 2. The GP s expected payoff by using the socially optimal contract, Cs = (K = 2I,l = 1,s s), where s s solves problem (P), equals E[s s(x)] = min[e[s o (x)],π s ], (10) where E[s o (x)] is presented in Proposition 1 and π s is given by equation (2). 3.3 LPs rents By part (i) in Proposition 1, when projects profitability potential is low, s o pledges all the marginal cash flows in the region [2I,2R] to the GP and gives the GP a payoff expressed in (5). Since this payoff is greater than the investment surplus πs, by Proposition 2, when projects have low profitability potential, s s, which is a call-option-like security with strike price equal to LPs committed capital, gives the GP all the investment surplus and thus leaves LPs with no rents. However, when projects profitability potential is medium or high, to satisfy the incentive-compatibility condition (IC BB ), s o must leave some marginal cash flows in 13

15 Low Profitability Potential Medium or High Profitability Potential (in the case where λ>λ) s s (x) s s (x) G P G P P A Y O F F 0 2I R R+I 2R x P A Y O F F 0 2I R R+I 2R x Cash Flow Cash Flow Panel A. R/I 3 2p Panel B. R/I > 3 2p and λ > λ s s (x) Medium Profitability Potential (in the case where λ λ) s s (x) High Profitability Potential (in the case where λ λ) G P G P P A Y O F F 0 2I R R+I 2R x P A Y O F F 0 2I R R+I 2R x Cash Flow Cash Flow Panel C. 3 2p < R/I 3 and λ λ Panel D. R/I > 3 and λ λ Figure 2: The GP security, s s, that gives the GP the highest payoff among all the socially optimal securities. The graphs illustrate the shapes of s s in four cases, distinguished by parametric assumptions. Panel A illustrates the case where projects profitability potential, represented by R/I, is low (below 3 2p). Panel B illustrates the case where projects profitability potential is medium or high (above 3 2p) while a project s probability of being good is high, i.e., λ > λ. Panel C illustrates the case where projects profitability potential is medium (between 3 2p and 3) while a project s probability of being good is low, i.e., λ λ. Panel D illustrates the case where projects profitability potential is high (above 3) while a project s probability of being good is low, i.e., λ λ. the region [2I,2R] to LPs. Note that these marginal cash flows are pledged to LPs not because of LPs break-even condition but because of (IC BB ). Thus, it is possible that, at the solution to the primal security design problem (P), condition (IC BB ) is binding while LPs break-even condition is not. In this case, LPs earn positive rents. We use the case stated in part (iiia) in Proposition 1 as an example to show that LPs may earn rents when the projects s profitability potential is not too low. In part (iiia), projects have high profitability potential while low probability of having good quality. In this case, as shown in (iiia), regardless of the final cash flow, the GP s payoff cannot 14

16 exceed 2I. Since the GP s expected payoff equals the minimum of E[s o (x)] and the investment surplus, π s, and since, by equation (9), E[s o (x)] does not depend on R, an increase in R only increases π s but not E[s o (x)]. When R is sufficiently large, π s exceeds E[s o (x)], in which case, by using a socially optimal contract, the GP earns at most E[s o (x)], leaving π s E[s o (x)] as rents conceded to LPs. Note that, in practice, good projects are typically a minority of a PE fund s project candidates while PE-backed projects, conditioned on success, often produce stellar profits; examples include Apple, Federal Express, Genentech, Intel, Microsoft, etc. Thus, the parametric conditions in (iiia) seem to be consistent with the PE practice. Thus, the result in (iiia) suggests that the socially optimal contract can be costly for the GP to use because of rent concession. The rents paid to LPs can be considerably high if the projects profitability potential is sufficiently high. For instance, suppose p = 0.1, λ = 0.2, R = 8, and I = 1. These parameters satisfy the conditions in (iiia). Given these parameters, by (9), E[s o (x)] = 0.848, while by (2), π s = Thus, the GP s payoff, min[π s,e[s o (x)]], equals 0.848, while the LPs rents equal = In this case, although the capital market is perfectly competitive, by using the socially optimal contract, the LPs payoff is more than twice as much as the GP s payoff and the LPs net expected rate of return, which equals 1.824/2, is more than 90%! It is a general result that, ceteris paribus, an increase in R (and hence an increase in projects profitability potential) weakly increases LPs rents. We enclose this result in the next proposition. Proposition 3. Fixing p, λ, and I, increasing R weakly increases LPs rents. Given that the GP may not earn all the investment surplus by using the socially optimal contract, we still need to determine whether the socially optimal contract offers the GP the highest expected payoff compared to less efficient but possibly rent-saving contracts. We tackle this problem in the next section by examining those alternative financing arrangements. 4 Alternative financing arrangements Recall from the discussion in Section 3 that there are four conditions for socially optimal contracting: (i) unconstrained capital, i.e., K = 2I, (ii) cross-pledging, (iii) short investment period, i.e., l = 1, and (iv) the incentive-compatibility condition (IC BB ). By relaxing one or more conditions, we eventually obtain four mutually exclusive alterna- 15

17 tive financing arrangements, listed as follows: 11 a. the capital-constrained arrangement, under which the GP establishes a small fund, i.e., K = I; 12 b. the long duration arrangement, under which the GP establishes a large fund which specifies a long investment period, i.e., K = 2I and l = 2; c. the stand-alone arrangement, under which the GP establishes two small funds, each with K = I; d. the socially suboptimal short duration arrangement, under which the GP establishes a large fund which specifies a short investment period, i.e., K = 2I and l = 1, but the GP security violates (IC BB ). In what follows, we analyze each of these alternative financing arrangements. 4.1 The capital-constrained arrangement By using the capital-constrained arrangement, the GP can at most undertake one project. Because of condition (NP), the GP can only obtain a positive payoff if he creates value for the fund. Thus, the GP always has an incentive to undertake one project. In addition, in state GB, given that the good project has a higher probability of success than the bad one, the GP always picks the good project. Therefore, the GP s investment behavior by using the capital-constrained arrangement is as follows: (i) in state GG, undertake one good project, (ii) in state GB, undertake one good project, and (iii) in state BB, undertake one bad project. It is thus clear that the capital-constrained arrangement minimizes overinvestment but causes underinvestment. The next proposition shows that the capital-constrained arrangement is feasible so long as the investment surplus from the prescribed investment behavior is nonnegative, and by using a call-option-like GP security under the capital-constrained arrangement, the GP concedes no rents to the LPs. Proposition 4. The investment surplus by using the capital-constrained arrangement is π c = ( 2λ λ 2 + (1 λ) 2 p ) R I. (11) The capital-constrained arrangement is feasible if and only if the surplus in (11) is nonnegative. Under the capital-constrained arrangement, the optimal GP security is 11 Renegotiation at time 2 cannot alleviate the overinvestment problem. This is because, at time 2, LPs cannot observe the progress of time 1 s investments and thus the ex post renegotiation is still subject to the problem of fly-by-night operators. As a result, any ex post contract that gives the GP a positive payoff without bringing success to any project will not be accepted by LPs. 12 With a small fund, the choice of the length of the investment period has no effect on investment efficiency. 16

18 given by s 0 if x I c(x) =, (12) α c (x I) if x > I ( ) where α c = R I 1 I R. By using this security under the capital-constrained 2λ λ 2 +(1 λ) 2 p arrangement, the GP concedes no rents to the LPs and captures all the investment surplus π c. 4.2 The long duration arrangement By using the long duration arrangement, the GP can at most undertake two projects. As discussed in Section 3.1, with a long investment period, in state BB, the GP has an incentive to invest in one bad project at time 1 and, conditioned on anticipating the failure of this investment at time 2, the GP has an incentive to invest in the other bad project at time 2. Thus, to minimize agency conflicts conditioned on using the long duration arrangement, the GP security must induce the GP to (i) undertake two good projects in state GG, (ii) undertake only the good project in state GB, and (iii) in state BB, conditioned on foreseeing at time 2 the success of his earlier investment, keep the dry powder in the safe asset. Among the three requirements, (i) is implied by the monotonicity condition (M), while (ii) and (iii) are satisfied if and only if the GP security satisfies the weaker incentive-compatibility condition, (IC GB ), presented in Section 3.1. The next proposition shows that the long duration arrangement is feasible so long as the investment surplus from the prescribed investment behavior is nonnegative, and by using a call-option-like GP security under the long duration arrangement, the GP concedes no rents to the LPs. Proposition 5. The investment surplus by using the long duration arrangement is π l = ( 2λ + (1 λ) 2 (2 p)p ) R ( 1 + λ 2 + (1 λ) 2 (1 p) ) I. (13) The long duration arrangement is feasible if and only if the surplus in (13) is nonnegative. Under the long duration arrangement, the optimal GP security is given by s l (x) = 0 if x 2I, (14) α l (x 2I) if x > 2I where α l = (2λ+(1 λ)2 (2 p)p)r (1+λ 2 +(1 λ) 2 (1 p))i. By using this security under the (2λ+(1 λ) 2 (2 p)p)r (2λ+(1 λ) 2 (3 2p)p)I long duration arrangement, the GP concedes no rents to the LPs and captures all the 17

19 investment surplus π l. 4.3 The stand-alone and the socially suboptimal short duration arrangement Under the stand-alone arrangement, the GP finances two projects by establishing two small funds, with each fund financing one project. Since the GP security in each fund must satisfy condition (NP), the GP has an incentive to undertake both projects in any state, which implies that the GP undertakes two bad projects in state BB. Under the socially suboptimal short duration arrangement, the GP security violates condition (IC BB ), in which case, given the short investment period, the GP also undertakes two bad projects in state BB. Since the long duration arrangement only causes inefficiency in state BB, in which case the GP invests in 2 p bad projects on average, it is clear that the long duration arrangement is strictly more efficient than both the stand-alone and the socially suboptimal short duration arrangement. Moreover, by Proposition 5, by using the long duration arrangement, the GP concedes no rents to the LPs. The next proposition is thus straightforward. Proposition 6. The GP s payoff is higher under the long duration arrangement than under the stand-alone and the socially suboptimal short duration arrangement. 5 The equilibrium financing arrangement Proposition 6 implies that there are three financing arrangement candidates for the equilibrium: the socially optimal arrangement, the capital-constrained arrangement, and the long duration arrangement. Each has its own pros and cons. The socially optimal arrangement maximizes efficiency but may force the GP to concede rents to the LPs. As discussed in Section 3.3, rent concession occurs only if project profitability potential is not too low. Thus, the socially optimal arrangement will be used if project profitability potential is sufficiently low. When project profitability potential is not too low, the socially optimal arrangement causes rent concession, in which case the GP may turn to the capital-constrained or the long duration arrangement. Both these two alternative arrangements are rent-saving, but the capital-constrained arrangement causes underinvestment while the long duration arrangement causes more overinvestment. Everything else being equal, the higher the projects profitability potential, the larger the efficiency loss induced by underinvestment while the smaller the efficiency loss induced by overinvestment. Thus, the capital-constrained arrangement will be used only if project prof- 18

20 itability potential is neither too low nor too high, whereas the long duration arrangement will be used only if project profitability potential is sufficiently high. We present these results formally in the following theorem. Theorem 1. Suppose condition (3) is satisfied so that financing is feasible. The equilibrium financing arrangement has the following features: i. if R/I 3 2p, the GP chooses the socially optimal arrangement; ii. fixing p and λ, as R/I 1/p, the GP either chooses the socially optimal arrangement or the long duration arrangement; iii. fixing p and λ, if the GP chooses the socially optimal arrangement as R/I 1/p, then he chooses the socially optimal arrangement for all R/I < 1/p; iv. fixing p and λ, if the GP chooses the long duration arrangement as R/I 1/p, then there must either (a) exist a unique threshold r sl (3 2p,1/p) such that the GP chooses the socially optimal arrangement when R/I r sl and chooses the long duration arrangement when r sl < R/I < 1/p or (b) exist two thresholds r sc and r cl, where 3 2p < r sc < r cl < 1/p, such that the GP chooses the socially optimal arrangement when R/I r sc, chooses the capital-constrained arrangement when r sc < R/I r cl, and chooses the long duration arrangement when r cl < R/I < 1/p. Theorem 1 implies that the lower (higher) the projects profitability potential, the more likely the use of the socially optimal arrangement (the long duration arrangement) and the capital-constrained arrangement is used only if project profitability potential is moderate. The result thus suggests a positive relation between project profitability potential and the length of the investment period and a U-shaped relation between project profitability potential and fund size. We use an example to show how financing arrangement changes when projects profitability potential changes. In this example, p = 0.1 and λ = 0.2. We change projects profitability potential by changing R/I. By assumption, 2 < R/I < 1/p. Thus, given that p = 0.1, we must have 2 < R/I < 10. Given these parametric assumptions, there exist two thresholds π sc and π cl such that the socially optimal arrangement is used if R/I falls below π sc, the capital-constrained arrangement is used if R/I is between π sc and π cl, while the long duration arrangement is used if R/I is above π cl. 19

21 6 Implications of the model Our model contains several implications on PE fund structures and a number of predictions for GP behavior, PE fund size and performance, and LP earnings. Some of these predictions are consistent with the existing empirical findings, while others are potentially testable in future research. Below we talk in detail about model implications from several aspects. 6.1 Length of the investment period Our model suggests that the length of the investment period can potentially vary with the project characteristics. However, given the fact that PE funds typically invest in projects with high profitability potential, our result implies that PE funds may contractually define a relatively long investment period in practice. This is consistent with the evidence in Giot, Hege, and Schwienbacher (2014) that, although PE funds usually specify the first 5 years as the investment period, about 60% of all PE investments are made within the first 2 years and about 80% within the first 3 years, which suggests that PE funds are able to build their portfolio relatively quickly and the 5-year constraint does not seem to be binding in most cases. 6.2 Capping late investments Degeorge, Martin, and Phalippou (2013) find that the SBOs made in the late phase of the investment period of a PE fund not only underperform the SBOs made in the early phase but have negative NPVs on average. They thus suggest that LPs might benefit from capping the percentage of a fund that the GP may invest in late SBOs. The effect of capping the late investments is similar to the effect of shortening the investment period in our model: they both restrict the GP s ability to base investment decisions on the fund s cumulative performance. To evaluate its effect on investment efficiency, we have to take into account both its direct effect on the GP s late investment behavior and its indirect effect on the GP s early investment behavior. Our result suggests that, if a PE fund caps the late investments without sufficiently capping the GP s upside sharing, the agency problem can be exacerbated: knowing that the late investments are capped, the GP, with a considerable share in the upside, bets more heavily on the upside in the early phase of the investment period and he may undertake more bad deals in total. Thus, to design the socially optimal contract, GP compensation and late investment restriction must be considered jointly and, when project profitability potential is sufficiently large, concavifying GP compensation by, for example, introducing a GP s Carried Interest 20